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Apart from checking options and approximation, it can also be solved using factorization of the expression:

\(n^3 - n^2\) = 180
Or, \(n^3 - n^2\) - 180 = 0
Or, \(n^3 - 6n^2 + 5n^2\) - 30n + 30n - 180 = 0
Or, \(n^2\) (n - 6) + 5n (n - 6) + 30 (n - 6) = 0
Or, (n - 6) (\(n^2\)+ 5n + 30) = 0
from here only, we can say n = 6, as \(n^2\) + 5n + 30 is always > 0 for any positive value of n

Hence, correct answer is Option E
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Bunuel
If the cube of n is 180 greater than the square of n, then n =

(A) 10
(B) 9
(C) 8
(D) 7
(E) 6


Though I totally messed his question, I would try to add my learning.

n^3 = n^2 + 180

n^3 - n^2 = 180 (Don't try to make a polynomial from here)

Now try answer choices one by one:

A. 10^3 - 10^12 = 180

900 = 180 (Incorrect)

E. 6^3 - 6^2 = 180

216 - 36 = 180

180 = 180 (Correct)

Hence (E)
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Bunuel
If the cube of n is 180 greater than the square of n, then n =

(A) 10
(B) 9
(C) 8
(D) 7
(E) 6

We can create the equation:

n^3 = 180 + n^2

n^3 - n^2 - 180 = 0

We can see that it’s not easy to factor n^3 - n^2 - 180. Therefore, we will just check the answer choices and see which one satisfies the equation. Notice the cube root of 180 is less than 6 (because 6^3 = 216), so we’ll try the answer choices in reverse. That is, we’ll try answer choice E first.

If n = 6, then 6^3 - 6^2 - 180 = 216. - 36 - 180 = 0.

We see that we’ve found our answer.

Answer: E
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Bunuel
If the cube of n is 180 greater than the square of n, then n =

(A) 10
(B) 9
(C) 8
(D) 7
(E) 6

Given n^3 = 180 + n^2
—> n^3 - n^2 = 180
—> n^2(n - 1) = 180 = 36*5
—> n^2*(n - 1) = 6^2*(6 - 1)

So, n = 6.

IMO Option E

Pls Hit kudos if you like the solution

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